Hi Rory, in keeping with your findings, the 3V3 appears to be regulated independently in most ATX designs. I'm not encouraging anyone to open up the PSU unless they know exactly what they're doing so I've been restricting myself to probing around on the outside. To my way of thinking external corrections are the only ones really worth considering. The 12V system on the Alpine does at least appear to have some feedback but moves with the 5V. All I've had time to do is confirm that extra loading on the 5V increases the voltage on the 12V as well. So it may be possible to squeeze something approaching an extra Volt right where it's needed with a bigger dummy load.
I had previously rolled dc42's suggestion of using the external Bed heater MOSFET drive into a crude voltage stabilisation scheme but it's an unnecessary elaboration given that everything running from 12V is unfussy about the actual voltage - only the heaters which require a minimum voltage in order to attain the temperatures required. Because of this, for maximum simplicity, the 5V could be permanently loaded with something more useful - like dc's print bed illumination. Unfortunately the cost of "whatever you hang on the 5V" seems to outweigh this approach against the additional cost of sourcing a better PSU unless the load has a particular use. It's a tough one.
I think the only other runner is to boost the 12V by a volt or two using a PWM switcher using a circuit similar to
this random design
Higher output current can be obtained with the use of an intermediate mosfet driver and better mosfet along with a higher switching frequency, lower value of inductance and uprated diode etc. Most of that only represents a few Euros except for the inductor which would have to be a bit of a beast. Something like the ones in the output of a typical ATX PSU
RS Components Reprap Ormerod No. 481